Do neurons of a neural network model a linear relationship?
In a neural network (NN), a neuron can act as a linear operator, but it usually acts as a non-linear one. The usual equation of a neuron $i$ in layer $l$ of an NN is 

$$o_i^l = \sigma(\mathbf{x}_i^l \cdot  \mathbf{w}_i^l + b_i^l),$$ 

where $\sigma$ is a so-called activation function, which is usually a non-linearity, but it can also be the identity function, $\mathbf{x}_i^l$ and $\mathbf{w}_i^l$ are the vectors that respectively contain the inputs and the weights for neuron $i$ in layer $l$, and $b_i^l \in \mathbb{R}$ is a bias. Similarly, the output of a layer of a feed-forward neural network (FFNN) is computed as

$$\mathbf{o}^l = \sigma(\mathbf{X}^l \mathbf{W}^l + \mathbf{b}^l).$$

In your specific example, you set the new weight to $0$, if the output of the linear combination is less than $0$, else you use the output of the linear combination. This is the definition of the ReLU activation function, which is a non-linear function.